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# Mathematics and Logic - Skill and Concept Development

with lessons and lesson ideas at many levels. If one site element is not to your liking, try another. Each one is different.

Online Volumes: 1 Elements of Reason || 2 Three Skills For Algebra || 3 Why Slopes Light Calculus Preview or Intro plus Hard Calculus Proofs, decimal-based.
More Lessons &Lesson Ideas: Arithmetic & No. Theory || Time & Date Matters || Algebra Starter Lessons || Geometry - maps, plans, diagrams, complex numbers, trig., & vectors || More Algebra || More Calculus || DC Electric Circuits || 1995-2011 Site Title: Appetizers and Lessons for Mathematics and Reason

Mathematics Concept & Skill Development Lecture Series: Webvideo consolidation of site lessons and lesson ideas in preparation. Price to be determined.

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Are you a careful reader, writer and thinker? Five logic chapters lead to greater precision and comprehension in reading and writing at home, in school, at work and in mathematics.
- 1 versus 2-way implication rules - A different starting point - Writing or introducting the 1-way implication rule IF B THEN A as A IF B may emphasize the difference between it or the latter, and the 2-way implication A IF and ONLY IF B.
- Deductive Chains of Reason - See which implications can and cannot be used together to arrive at more implications or conclusions,
- Mathematical Induction - a light romantic view that becomes serious.
- Responsibility Arguments - his, hers or no one's
- Islands and Divisions of Knowledge - a model for many arts and disciplines including mathematics course design: Different entry points may make learning and teaching easier. Are you ready for them?

#### Early High School Arithmetic

Deciml Place Value - funny ways to read multidigit decimals forwards and backwards in groups of 3 or 6.
- Decimals for Tutors - lean how to explain or justify operations. Long division of polynomials is easier for student who master long division with decimals.
- Primes Factors - Efficient fraction skills and later studies of polynomials depend on this.
- Fractions + Ratios - See how raising terms to obtain equivalent fractions leads to methods for addition, comparison, subtraction, multiplication and division of fractions.
- Arithmetic with units - Skills of value in daily life and in the further study of rates, proportionality constants and computations in science & technology.

#### Early High School Algebra

What is a Variable? - this entertaining oral & geometric view may be before and besides more formal definitions - is the view mathematically correct?
- Formula Evaluation - Seeing and showing how to do and record steps or intermediate results of multistep methods allows the steps or results to be seen and checked as done or later; and will improve both marks and skill. The format here allows the domino effects of care and the domino effects of mistakes to be seen. It also emphasizes a proper use of the equal sign.
- Solve Linear Eqns with & then without fractional operations on line segments - meet an visual introduction and learn how to present do and record steps in a way that demonstrate skill; learn how to check answers, set the stage for solving word problems by by learning how to solve systems of equations in essentially one unknown, set the stage for solving triangular and general systems of equations algebraically.
- Function notation for Computation Rules - another way of looking at formulas. Does a computation rule, and any rule equivalent to it, define a function?
- Axioms [some] as equivalent Computation Rule view - another way for understanding and explaining axioms.
- Using Formulas Backwards - Most rules, formulas and relations may be used forwards and backwards. Talking about it should lead everyone to expect a backward use alone or plural, after mastery of forward use. Proportionality relations may be use backward first to find a proportionality constant before being used forwards and backwards to solve a problem.

#### Early High School Geometry

Maps + Plans Use - Measurement use maps, plans and diagrams drawn to scale.
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- Coordinates - Use them not only for locating points but also for rotating and translating in the plane.
- What is Similarity - another view of using maps, plans and diagrams drawn to scale in the plane and space. Many human-made objects are similar by design.
- 7 Complex Numbers Appetizer. What is or where is the square root of -1. With rectangular and polar coordinates, see how to add, multiply and reflect points or arrows in the plane. The visual or geometric approach here known in various forms since the 1840s, demystifies the square root of -1 and the associated concept of "imaginary" numbers. Here complex number multiplication illustrates rotation and dilation operations in the plane.
- Geometric Notions with Ruler & Compass Constructions :
1 Initial Concepts & Terms
2 Angle, Vertex & Side Correspondence in Triangles
3 Triangle Isometry/Congruence
4 Side Side Side Method
5 Side Angle Side Method
6 Angle Bisection
7 Angle Side Angle Method
8 Isoceles Triangles
9 Line Segment Bisection
10 From point to line, Drop Perpendicular
11 How Side Side Side Fails
12 How Side Angle Side Fails
13 How Angle Side Angle Fails

www.whyslopes.com >> - Volume 1A Pattern Based Reason >> Postscript B More-on-Story-Telling-and-Reason Next: [Postscript C Consistency-as-a-Tool-for-Reason.] Previous: [Postscript A Story-Telling.]   [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28][29] [30]

## Appendix B. More on Story Telling and Reason

November 2010

Human kind has been telling stories and drawing pictures for thousands of years. Some stories and pictures reflect reality or observations well. Other stories and pictures may be works of imagination stemming from dreams or done for entertainment or given for self-protection. Stories and pictures do not have to tell the truth in order to be followed and understood. Plays and stories may provide a virtual reality with rules and patterns that may be ordinary or not.

The ability to follow the steps or actions in story, in sequence and in parallel, sets the stage for the ability to give and follow instructions and theories step by step. Dreams and nightmares recounted as stories need not be consistent with our experience of reality. In them, inconsistent and mutually exclusive events may be seen and imagined. In leaving childhood, one task is to distinguish between be awake and dreaming. However, under longterm stress or sleeplessness, we may lose our sense of the boundary between reality and imagination.

Rules and patterns given to describe reality may be used alone or in combination. They may be approximate - true in some circumstances but not all. But in chaining them together, using them directly or in contrapositive form we may arrive at a theory of what is to tell and check. Just a for moment, imagine that we want to compose or extend a story that resembles reality, or what we hope reality is.

Once a story is partly written, we may want to extend it. In the extension, we might have a situation A or NOT A occur. The occurrence of the situation A is a possible addition to the story if it does not imply an immediate contradiction or inconsistency with the earlier part of the story. The non-occurrence of the situation A is a possible addition to the story if it does not imply an inconsistency with an earlier part of the story. Here the story teller could say the situation A occurred and that would give one extension, or (s)he could say that the situation did not occur for another extension. But then the story teller may decide to extend the story without a mention of the occurrence or not of the situation A. So in story telling, we have an extension in which both A and NOT A do not occur in the story, and neither has too. So the law of excluded middle, the statement that

A or NOT A must hold

is not true for story telling - a story composer (canonical or not) does not have to choose A or not A. Many virtual realities are thus possible in story telling. No connection to the five senses is required. That makes fiction possible.

Proof by contradiction may be related to extending a story or theory by requiring consistency.

• the situation A has to be part of a story or theory if its non-occurrence makes the story inconsistent. Whether or not the occurrence of A is consistent with the story or theory is another question, and likewise,
• the situation not A (meaning the non-occurrence of A) has to be part of the story or theory if the occurrence of A makes the story inconsistent. Whether or not the non-occurrence of A is consistent with the story or another is another question.

Both stories and theories may grow in tree or branch like manner, with a shoot here, a shoot there and shoot over-there. In growing a story or developing a theory by adding new premises or by following chains of reason based on earlier premises and earlier chains of reason, we may not know what is around the corner. We may not be certain the story or its theory with all implications and premises is consistent. But we follow the ideas and steps in a story, theory or subtheory where that is practical, and empirically learn where, but may not know in advance if the story and theory as is or extended is consistent in that contradictions or mutually exclusive ideas are avoided, or is at least consistent with reality in those cases where the story or theory fits close enough to be useful. We may judge stories, theories and practices (patterns and detective work included) by their internal consistency and by their consistency with external factors. Competing stories or theories may exist and be useful side by side.

In the physical sciences, different mechanisms to classify ionic substances as acid, base or salt may work and agree in many situations, but they disagree in some. So substances which should be salts in accordance with a theory based on their chemical composition may actually be acidic or basic (slightly and not strongly) in accordance with observations based on litmus paper or pH-meters.

Remark: This chapter and the addendum reflect the question of what to assume and why in mathematics and also in logic, what methods for arriving at conclusions to assume or employ beyond the direct use of an implication and its contrapositive. To learn more about what might be possible, one may study mathematical logic. But in that, the conclusion that not all is certain represents a loss of certainty, one spoken about in a book

Mathematics: The loss of certainty, Morris Kline, Oxford University Press, 1980-2

one that may be offset by learning about rules and patterns, their origins, where they work and their limitations. That represent a viable avenue in some fields of endeavour. But the loss of certainty is a disappointment for students who study mathematics and logic because of the apparent power and certainty of its methods in itself and in many fields of endeavour. But that power and that certain naively seen are finite, and so leave room for thought. Enough said. Good luck.

Selby A, Volume 1A, Pattern Based Reason, 1996.

www.whyslopes.com >> - Volume 1A Pattern Based Reason >> Postscript B More-on-Story-Telling-and-Reason Next: [Postscript C Consistency-as-a-Tool-for-Reason.] Previous: [Postscript A Story-Telling.]   [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28][29] [30]

Road Safety Messages for All: When walking on a road, when is it safer to be on the side allowing one to see oncoming traffic?

Play with this [unsigned] Complex Number Java Applet to visually do complex number arithmetic with polar and Cartesian coordinates and with the head-to-tail addition of arrows in the plane. Click and drag complex numbers A and B to change their locations.

#### Pattern Based Reason

Online Volume 1A, Pattern Based Reason, describes origins, benefits and limits of rule- and pattern-based reason and decisions in society, science, technology, engineering and mathematics. Not all is certain. We may strive for objectivity, but not reach it. Online postscripts offer a story-telling view of learning: [ A ] [ B ] [ C ] [ D ] to suggest how we share theory and practice in many fields of knowledge.

#### Site Reviews

1996 - Magellan, the McKinley Internet Directory:

Mathphobics, this site may ease your fears of the subject, perhaps even help you enjoy it. The tone of the little lessons and "appetizers" on math and logic is unintimidating, sometimes funny and very clear. There are a number of different angles offered, and you do not need to follow any linear lesson plan. Just pick and peck. The site also offers some reflections on teaching, so that teachers can not only use the site as part of their lesson, but also learn from it.

2000 - Waterboro Public Library, home schooling section:

CRITICAL THINKING AND LOGIC ... Articles and sections on topics such as how (and why) to learn mathematics in school; pattern-based reason; finding a number; solving linear equations; painless theorem proving; algebra and beyond; and complex numbers, trigonometry, and vectors. Also section on helping your child learn ... . Lots more!

2001 - Math Forum News Letter 14,

... new sections on Complex Numbers and the Distributive Law for Complex Numbers offer a short way to reach and explain: trigonometry, the Pythagorean theorem,trig formulas for dot- and cross-products, the cosine law,a converse to the Pythagorean Theorem

2002 - NSDL Scout Report for Mathematics, Engineering, Technology -- Volume 1, Number 8

Math resources for both students and teachers are given on this site, spanning the general topics of arithmetic, logic, algebra, calculus, complex numbers, and Euclidean geometry. Lessons and how-tos with clear descriptions of many important concepts provide a good foundation for high school and college level mathematics. There are sample problems that can help students prepare for exams, or teachers can make their own assignments based on the problems. Everything presented on the site is not only educational, but interesting as well. There is certainly plenty of material; however, it is somewhat poorly organized. This does not take away from the quality of the information, though.
... section Solving Linear Equations ... offers lesson ideas for teaching linear equations in high school or college. The approach uses stick diagrams to solve linear equations because they "provide a concrete or visual context for many of the rules or patterns for solving equations, a context that may develop equation solving skills and confidence." The idea is to build up student confidence in problem solving before presenting any formal algebraic statement of the rule and patterns for solving equations. ...

#### Senior High School Geometry

- Euclidean Geometry - See how chains of reason appears in and besides geometric constructions.
- Complex Numbers - Learn how rectangular and polar coordinates may be used for adding, multiplying and reflecting points in the plane, in a manner known since the 1840s for representing and demystifying "imaginary" numbers, and in a manner that provides a quicker, mathematically correct, path for defining "circular" trigonometric functions for all angles, not just acute ones, and easily obtaining their properties. Students of vectors in the plane may appreciate the complex number development of trig-formulas for dot- and cross-products.
Lines-Slopes [I] - Take I & take II respectively assume no knowledge and some knowledge of the tangent function in trigonometry.

#### Calculus Starter Lessons

Why study slopes - this fall 1983 calculus appetizer shone in many classes at the start of calculus. It could also be given after the intro of slopes to introduce function maxima and minima at the ends of closed intervals.
- Why Factor Polynomials - Online Chapter 2 to 7 offer a light introduction function maxima and minima while indicating why we calculate derivatives or slopes to linear and nonlinear curves y =f(x)
- Arithmetic Exercises with hints of algebra. - Answers are given. If there are many differences between your answers and those online, hire a tutor, one has done very well in a full year of calculus to correct your work. You may be worse than you think.